# Turbulence via Intermolecular Potential: Viscosity and Transition Range of the Reynolds Number

## Abstract

**:**

## 1. Introduction

## 2. Our Model of Inertial Turbulent Gas Flow

## 3. A Model of Inertial Turbulent Gas Flow with Viscosity

## 4. Numerical Simulations

#### Results

## 5. Summary

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Boussinesq, J. Essai sur la Théorie des Eaux Courantes; Imprimerie Nationale: Paris, France, 1877. [Google Scholar]
- Thomson, W. XLV. On the Propagation of Laminar Motion through a Turbulently Moving Inviscid Liquid. Philos. Mag. Ser.
**1887**, 24, 342–353. [Google Scholar] [CrossRef] [Green Version] - Reynolds, O. III. An Experimental Investigation of the Circumstances which Determine whether the Motion of Water shall be Direct or Sinuous, and of the Law of Resistance in Parallel Channels. Proc. R. Soc. Lond.
**1883**, 35, 84–99. [Google Scholar] [CrossRef] [Green Version] - Kolmogorov, A. The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers. Dokl. Akad. Nauk SSSR
**1941**, 30, 299–303. [Google Scholar] [CrossRef] - Kolmogorov, A. On Degeneration of Isotropic Turbulence in an Incompressible Viscous Liquid. Dokl. Akad. Nauk SSSR
**1941**, 31, 538–541. [Google Scholar] - Kolmogorov, A. Dissipation of Energy in the Locally Isotropic Turbulence. Dokl. Akad. Nauk SSSR
**1941**, 32, 19–21. [Google Scholar] [CrossRef] - Reynolds, O. IV. On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion. Phil. Trans. Roy. Soc. A
**1895**, 186, 123–164. [Google Scholar] [CrossRef] - Richardson, L. Atmospheric Diffusion Shown on a Distance-Neighbour Graph. Proc. Roy. Soc. Lond. A
**1926**, 110, 709–737. [Google Scholar] [CrossRef] [Green Version] - Taylor, G. Statistical Theory of Turbulence. Proc. Roy. Soc. Lond. A
**1935**, 151, 421–444. [Google Scholar] [CrossRef] [Green Version] - Taylor, G. The Spectrum of Turbulence. Proc. Roy. Soc. Lond. A
**1938**, 164, 476–490. [Google Scholar] [CrossRef] [Green Version] - de Kármán, T.; Howarth, L. On the Statistical Theory of Isotropic Turbulence. Proc. Roy. Soc. Lond. A
**1938**, 164, 192–215. [Google Scholar] [CrossRef] - Prandtl, L. Beitrag zum Turbulenzsymposium. In Proceedings of the Fifth International Congress on Applied Mechanics, Cambridge, MA, USA, 12–16 September 1938; Den Hartog, J., Peters, H., Eds.; John Wiley: New York, NY, USA, 1938; pp. 340–346. [Google Scholar]
- Obukhov, A. On the Distribution of Energy in the Spectrum of a Turbulent Flow. Bull. Acad. Sci. USSR Geog. Geophys.
**1941**, 5, 453–466. [Google Scholar] - Obukhov, A. Structure of the Temperature Field in Turbulent Flow. Izv. Akad. Nauk SSSR Ser. Geogr. Geofiz.
**1949**, 13, 58–69. [Google Scholar] - Chandrasekhar, S. On Heisenberg’s Elementary Theory of Turbulence. Proc. Roy. Soc.
**1949**, 200, 20–33. [Google Scholar] [CrossRef] - Corrsin, S. On the Spectrum of Isotropic Temperature Fluctuations in an Isotropic Turbulence. J. Appl. Phys.
**1951**, 22, 469–473. [Google Scholar] [CrossRef] - Kolmogorov, A. A Refinement of Previous Hypotheses Concerning the Local Structure of Turbulence in a Viscous Incompressible Fluid at High Reynolds Number. J. Fluid Mech.
**1962**, 13, 82–85. [Google Scholar] [CrossRef] [Green Version] - Obukhov, A. Some Specific Features of Atmospheric Turbulence. J. Geophys. Res.
**1962**, 67, 3011–3014. [Google Scholar] [CrossRef] - Kraichnan, R. Dispersion of Particle Pairs in Homogeneous Turbulence. Phys. Fluids
**1966**, 9, 1937–1943. [Google Scholar] [CrossRef] - Kraichnan, R. Isotropic Turbulence and Inertial Range Structure. Phys. Fluids
**1966**, 9, 1728–1752. [Google Scholar] [CrossRef] - Saffman, P. The Large-Scale Structure of Homogeneous Turbulence. J. Fluid Mech.
**1967**, 27, 581–593. [Google Scholar] [CrossRef] [Green Version] - Saffman, P. A Model for Inhomogeneous Turbulent Flow. Proc. Roy. Soc. Lond. A
**1970**, 317, 417–433. [Google Scholar] [CrossRef] - Mandelbrot, B. Intermittent Turbulence in Self-Similar Cascades; Divergence of High Moments and Dimension of the Carrier. J. Fluid Mech.
**1974**, 62, 331–358. [Google Scholar] [CrossRef] - Avila, K.; Moxey, D.; de Lozar, A.; Avila, M.; Barkley, D.; Hof, B. The Onset of Turbulence in Pipe Flow. Science
**2011**, 333, 192–196. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Barkley, D.; Song, B.; Mukund, V.; Lemoult, G.; Avila, M.; Hof, B. The rise of fully turbulent flow. Nature
**2015**, 526, 550–553. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Khan, H.; Anwer, S.; Hasan, N.; Sanghi, S. Laminar to Turbulent Transition in a Finite Length Square Duct Subjected to Inlet Disturbance. Phys. Fluids
**2021**, 33, 065128. [Google Scholar] [CrossRef] - Vela-Martín, A. The Energy Cascade as the Origin of Intense Events in Small-Scale Turbulence. J. Fluid Mech.
**2022**, 937, A13. [Google Scholar] [CrossRef] - Wilcox, D. Turbulence Modeling for CFD, 2nd ed.; DCW Industries: La Cañada, CA, USA, 1998. [Google Scholar]
- Ferziger, J.; Perić, M. Computational Methods for Fluid Dynamics, 3rd ed.; Springer: Berlin/Heidelberg, Germany, 2002. [Google Scholar]
- Kays, W. Turbulent Prandtl Number—Where Are We? J. Heat Transf.
**1994**, 116, 284–295. [Google Scholar] [CrossRef] - Schmitt, F. About Boussinesq’s Turbulent Viscosity Hypothesis: Historical Remarks and a Direct Evaluation of its Validity. C. R. Mec.
**2007**, 335, 617–627. [Google Scholar] [CrossRef] [Green Version] - Abramov, R. Macroscopic Turbulent Flow via Hard Sphere Potential. AIP Adv.
**2021**, 11, 085210. [Google Scholar] [CrossRef] - Abramov, R. Turbulence in Large-Scale Two-Dimensional Balanced Hard Sphere Gas Flow. Atmosphere
**2021**, 12, 1520. [Google Scholar] [CrossRef] - Abramov, R. Creation of Turbulence in Polyatomic Gas Flow via an Intermolecular Potential. Phys. Rev. Fluids
**2022**, 7, 054605. [Google Scholar] [CrossRef] - Abramov, R. Turbulence via intermolecular potential: A weakly compressible model of gas flow at low Mach number. Phys. Fluids
**2022**, 34, 125104. [Google Scholar] [CrossRef] - Grad, H. On the Kinetic Theory of Rarefied Gases. Comm. Pure Appl. Math.
**1949**, 2, 331–407. [Google Scholar] [CrossRef] - Tsugé, S. Approach to the Origin of Turbulence on the Basis of Two-Point Kinetic Theory. Phys. Fluids
**1974**, 17, 22–33. [Google Scholar] [CrossRef] - Abramov, R. Turbulent Energy Spectrum via an Interaction Potential. J. Nonlinear Sci.
**2020**, 30, 3057–3087. [Google Scholar] [CrossRef] - Menon, E. Gas Pipeline Hydraulics; Taylor & Francis: Boca Raton, FL, USA, 2005. [Google Scholar]
- Letellier, C. Intermittency as a Transition to Turbulence in Pipes: A Long Tradition from Reynolds to the 21st century. C. R. Mec.
**2017**, 345, 642–659. [Google Scholar] [CrossRef] - Boublík, T. Hard-Sphere Radial Distribution Function from the Residual Chemical Potential. Mol. Phys.
**2006**, 104, 3425–3433. [Google Scholar] [CrossRef] - Chapman, S.; Cowling, T. The Mathematical Theory of Non-Uniform Gases, 3rd ed.; Cambridge Mathematical Library, Cambridge University Press: Cambridge, UK, 1991. [Google Scholar]
- Hirschfelder, J.; Curtiss, C.; Bird, R. The Molecular Theory of Gases and Liquids; Wiley: Hoboken, NJ, USA, 1964. [Google Scholar]
- Greenshields, C.; Weller, H.; Gasparini, L.; Reese, J. Implementation of Semi-Discrete, Non-Staggered Central Schemes in a Colocated, Polyhedral, Finite Volume Framework, for High-Speed Viscous Flows. Int. J. Numer. Methods Fluids
**2010**, 63, 1–21. [Google Scholar] [CrossRef] [Green Version] - Kurganov, A.; Tadmor, E. New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection–Diffusion Equations. J. Comput. Phys.
**2001**, 160, 241–282. [Google Scholar] [CrossRef] [Green Version] - van Leer, B. Towards the Ultimate Conservative Difference Scheme, II: Monotonicity and Conservation Combined in a Second Order Scheme. J. Comput. Phys.
**1974**, 14, 361–370. [Google Scholar] [CrossRef] - Weller, H.; Tabor, G.; Jasak, H.; Fureby, C. A Tensorial Approach to Computational Continuum Mechanics Using Object-Oriented Techniques. Comput. Phys.
**1998**, 12, 620–631. [Google Scholar] [CrossRef] - Buchhave, P.; Velte, C. Measurement of Turbulent Spatial Structure and Kinetic Energy Spectrum by Exact Temporal-to-Spatial Mapping. Phys. Fluids
**2017**, 29, 085109. [Google Scholar] [CrossRef] [Green Version] - Yu, S.T.; Tsai, Y.L.; Hsieh, K. Runge-Kutta Methods Combined with Compact Difference Schemes for the Unsteady Euler Equations. In Proceedings of the 28th Joint Propulsion Conference and Exhibit, Nashville, TN, USA, 6–8 July 1992; pp. 1–28. [Google Scholar] [CrossRef]

**Figure 1.**Longitudinal section of the computational domain. The domain dimensions are $36\times 5.2\times 5.2$ cm${}^{3}$. The inlet is on the left, and the outlet is on the right. The pipe walls are shown via thick black lines, so that both the inlet and outlet are visible. The boundary of the Fourier spectrum measurement region is shown in red. This region is a box of 24 cm in length and $3.6\times 3.6$ cm${}^{2}$ in cross-section.

**Figure 2.**Speed of the flow (m/s), expressed in the form of level curves, and captured in the longitudinal symmetry plane of the pipe at the elapsed time $t=0.15$ s for (

**a**) $\mathit{Re}=1000$, (

**b**) $\mathit{Re}=2000$, (

**c**) $\mathit{Re}=3000$, and (

**d**) $\mathit{Re}=4000$.

**Figure 3.**The Fourier spectra of the kinetic energy, averaged between $0.1$ and $0.2$ s of the elapsed time, for $\mathit{Re}=1000$, 2000, 3000, and 4000, as well as air (${\mu}_{0}=1.825\xb7{10}^{-5}$ kg/m s). The power decay slopes ${E}_{0}{k}_{x}^{-5/3}$ and ${E}_{0}{k}_{x}^{-8/3}$, with ${E}_{0}=20$ m${}^{2}$/s${}^{2}$, are added for reference.

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**MDPI and ACS Style**

Abramov, R.V.
Turbulence via Intermolecular Potential: Viscosity and Transition Range of the Reynolds Number. *Fluids* **2023**, *8*, 101.
https://doi.org/10.3390/fluids8030101

**AMA Style**

Abramov RV.
Turbulence via Intermolecular Potential: Viscosity and Transition Range of the Reynolds Number. *Fluids*. 2023; 8(3):101.
https://doi.org/10.3390/fluids8030101

**Chicago/Turabian Style**

Abramov, Rafail V.
2023. "Turbulence via Intermolecular Potential: Viscosity and Transition Range of the Reynolds Number" *Fluids* 8, no. 3: 101.
https://doi.org/10.3390/fluids8030101